小需求的多维资源调度

中文摘要 AI 辅助

我们研究多维资源调度问题。每个作业 $i$ 有一个 $d$ 维资源需求向量 $v_i$ 和一个处理时间 $s_i$。调度器为每个作业分配一个开始时间，约束条件是在每个时刻，正在处理的作业的总需求在任何资源维度上都不超过 $1$。目标是最小化完工时间（makespan）。我们关注每个个体资源需求都很小的情形。我们询问已知的多维向量装箱（对应单位处理时间的特例）中有利的\emph{小向量现象}是否扩展到具有异构处理时间的作业。关键难点在于任意处理时间会在多个持续时间尺度上产生时间交互：一个长作业可能与许多短作业重叠，而可行性必须在每个作业的执行区间内始终保持。我们证明了小向量现象在这种时间设置下依然存在。对于任意 $0<\epsilon<1/4$，在将最大处理时间归一化为 $1$ 后，如果每个需求向量的每个坐标至多为 $O(\epsilon^2/\log(d/\epsilon))$，我们给出一个随机离线算法，该算法产生的调度期望完工时间至多为 $(1+6\epsilon)\mathrm{OPT}+3$。因此，足够小的资源需求在任意维度下都允许渐近接近最优的调度，尽管处理时间异构。我们还在在线设置中获得了常数竞争比。设 $T$ 表示最大和最小处理时间之间的比率。如果每个坐标至多为 $O(1/(\log d\log T))$，我们给出一个随机 $O(1)$-竞争算法，其竞争比与 $d$ 和 $T$ 都无关。我们进一步对我们的方法进行去随机化，在可比的小性假设下获得确定性 $O(1)$-竞争算法。

英文摘要

We study multidimensional resource scheduling. Each job $i$ has a $d$-dimensional resource-demand vector $v_i$ and a processing time $s_i$. The scheduler assigns a start time to each job, subject to the constraint that, at every time, the total demand of the jobs being processed does not exceed $1$ in any resource dimension. The objective is to minimize the makespan. We focus on the regime in which every individual resource demand is small. We ask whether the favorable \emph{small-vector phenomenon} known for multidimensional vector packing, which corresponds to the special case of unit processing times, extends to jobs with heterogeneous processing times. The key difficulty is that arbitrary processing times create temporal interactions across multiple duration scales: a long job may overlap many shorter jobs, while feasibility must be maintained throughout every job's execution interval. We prove that the small-vector phenomenon persists in this temporal setting. For any $0<ε<1/4$, after normalizing the maximum processing time to $1$, if every coordinate of every demand vector is at most $O(ε^2/\log(d/ε))$, we give a randomized offline algorithm that produces a schedule with expected makespan at most $ (1+6ε)\mathrm{OPT}+3$. Thus, sufficiently small resource demands admit asymptotically near-optimal schedules in arbitrary dimension, despite heterogeneous processing times. We also obtain constant competitive ratios in the online setting. Let $T$ denote the ratio between the maximum and minimum processing times. If every coordinate is at most $O(1/(\log d\log T))$, we give a randomized $O(1)$-competitive algorithm, with a competitive ratio independent of both $d$ and $T$. We further derandomize our approach, obtaining a deterministic $O(1)$-competitive algorithm under a comparable smallness assumption.

发表机构

• Tel Aviv University（特拉维夫大学）
• University of Houston（休斯顿大学）

机构由 AI 辅助整理，请以论文原文为准。